Constrained Measurement Incompatibility from Generalised Contextuality of Steered Preparation

TL;DR

This paper explores the relationship between measurement incompatibility and generalized contextuality within general probabilistic theories, establishing new inequalities and constraints that deepen understanding of quantum and non-quantum theories.

Contribution

It introduces a novel inequality linking measurement compatibility and contextuality, and demonstrates how generalized contextuality constrains measurement incompatibility in GPTs.

Findings

Incompatibility of N measurements is necessary and sufficient for generalized contextuality.

Proposed inequalities are necessary conditions for N-wise compatibility.

Violations of inequalities quantify the degree of incompatibility.

Abstract

In a bipartite Bell scenario involving two local measurements per party and two outcome per measurement, the measurement incompatibility in one wing is both necessary and sufficient to reveal the nonlocality. However, such a one-to-one correspondence fails when one of the observers performs more than two measurements. In such a scenario, the measurement incompatibility is necessary but not sufficient to reveal the nonlocality. In this work, within the formalism of general probabilistic theory (GPT), we demonstrate that unlike the nonlocality, the incompatibility of N arbitrary measurements in one wing is both necessary and sufficient for revealing the generalised contextuality for the sub-system in the other wing. Further, we formulate a novel form of inequality for any GPT that are necessary for N-wise compatibility of N arbitrary observables. Moreover, we argue that any theory that violates the proposed inequality possess a degree of incompatibility that can be quantified through the amount of violation. Finally, we claim that it is the generalised contextuality that provides a restriction to the allowed degree of measurement incompatibility of any viable theory of nature and thereby super-select the the quantum theory.